Differential geometry of \(\mathfrak{g}\)-manifolds (Q1917361)

Let \(\mathfrak g\) be a finite-dimensional Lie algebra. A \(\mathfrak g\)-manifold is a manifold \(M\) with an action of \(\mathfrak g\), i.e. with an algebra homomorphism from \(\mathfrak g\) into the Lie algebra of all vector fields on \(M\). First of all, the authors consider the pseudogroup of all local transformations generated by an action of \(\mathfrak g\) and discuss its graph. If the orbits are equidimensional, many ideas from the theory of connections can be extended to the space of orbits. In such a situation, the authors study systematically the Frölicher-Nijenhuis bracket, curvature, covariant differentiation, Bianchi identity, parallel transport, basic differential forms, basic cohomology and characteristic classes.